Why is the average reward plot for my reinforcement learning agent different than the usual plots?
Well, the way to know that the agent is actually learning is by looking at its behavior while it performs the task, and by comparing against a known optimal performance.

So, does your agent reaches the goal quickly? Does it step out of the grid frequently? What is the maximum possible sum of rewards / minimum number of steps attainable? Is the agent close to that limit? From your graphic, and if I understood correctly your RL problem, the maximum average reward per step should be close to 1 (depending on the specific environment you are using), so I guess you are not so far from the optimal solution. 

Also, probably if you keep training for a longer period, your agent will reach a stable solution that might or might not be optimal. If you keep training after that, your curves surely will look like the ones you found online.